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A compact 9 point stencil based on integrated RBFs for the convection-diffusion equation. (English) Zbl 1427.65330

Summary: In this paper, a high-order compact stencil for solving the convection-diffusion equation in two dimensions is proposed. The convection and diffusion terms are both approximated by means of radial basis functions (RBFs) that are constructed over \(3\times 3\) rectangular stencils. Salient features here are that (i) integration is employed to construct local RBF approximations; and (ii) through the constants of integration, values of the convection-diffusion equation at several selected nodes on the stencil are also enforced. Numerical results indicate that (i) the inclusion of the governing equation into the stencil leads to a significant improvement in accuracy; (ii) when the convection dominates, accurate solutions are obtained at a regime of the RBF width which makes the RBFs peaked; and (iii) high levels of accuracy are achieved using relatively coarse grids.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
65D12 Numerical radial basis function approximation
76M99 Basic methods in fluid mechanics

Software:

Matlab; rbf_qr
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Full Text: DOI

References:

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